Math And Finance 1

Question I:

Let {X(t),t ≥ 0} be an arithmetic Brownian motion with a drift factor of 0.35 and a volatility of 0.43. Given that X4 = 2.

(a) Find the distribution of X(13).

(b) What is the probability that X(13) > 9.

(c) Find the confidence interval of X(13) at level 90%.

Question II:

(a) Show that the probability that a European call option will be exercised in a risk-neutral world, with the notation introduced in the BSM chapter, is equal to N(d2).

b) At time T, find an expression for the value of a derivative that pays off $100 if ST > K? (c) What is the value of this security at time zero using risk-neutral valuation.

2

Question III:

We assume that the stock price at time t, St, has a LogNormal model with S0 = 100, µ = 0.08 and σ = 0.3. It is assumed that the stock pays no dividend.

(a) Find P (S1 ≥ 105).

(b) Find P (S1 < 98). (c) Let Kt = S0ert, compute P (St ≥ Kt). Question IV: Find the price of a 3-month European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% and the volatility is 30% per annum.